I'm imagining six moons orbiting a single large planet along the same, circular path. Each moon is at the L5 Lagrange point of the one "ahead" of it, and at the L4 point of the one "behind". I realize this arrangement is unlikely to occur in nature, and that's ok. My question is, if it did somehow occur, is this a stable orbital configuration? If so, are there any restrictions that would apply? E.g., do the moons all need to be the same size? Can they spin, or no? What factors are likely to break this system?
The Lagrangian points are developed from the simple two-body problem treatment, in this case a moon (Moon 1) and a much more massive planet (Earth). As such anything in the Lagrangian 4 (L4)/Lagrangian 5 (L5) stable points would be need to be of negligible mass compared to Earth and Moon 1. Such a second object (Moon 2) in the L4/L5 would not have its own L4/L5 point unless it was massive enough that you could neglect the Moon 1's mass and treat the Moon 2 and Earth as a two-body problem, a contradiction.
Moon 2 can have a stable orbit as long as it lies outside of the sphere of influence of the Moon 1, i.e. far enough away that the effect of Moon 1's gravity is very small. By definition this means, it can not be in the L4/L5 points of Moon 1, since Moon 1's gravity defines all of the Lagrangian points.
To summarize six similar mass objects residing in the L4/L5 points of their neighbors is an unstable orbit. To study the orbits of multiple, similar sized objects you would need to use higher-order models, e.g. the three-body problem. You could get three moons, however, given a very massive planet, a massive prime moon, and two much smaller moons in the L4/L5 of the large one.
EDIT: Other options I thought of:
Six small moons could be in the same orbit, but depending on the size of your planet they might have to be too small for your purposes. Also a configuration like that is highly unlikely unless the moons were placed there or perhaps a larger moon was ripped apart by tidal effects (like Saturn's rings on a less dramatic scale).
There are other natural resonances between moons in different orbits that you could use. I don't remember exactly what they are but they're simple fractional relationships between the orbital periods.